Coupled Bending-Torsional Dynamic Behavior
of a Cantilever Beam Carrying Multiple Point
Masses
Alev Kacar Aksongur Faculty of Aeronautics and Astronautics
İstanbul Technical University İstanbul-Turkey
Email: [email protected]
Seher Eken Faculty of Aeronautics and Astronautics
Ondokuz Mayis University Samsun-Turkey
Email: [email protected]
Metin Orhan Kaya Faculty of Aeronautics and Astronautics
İstanbul Technical University İstanbul-Turkey
Email: [email protected]
Abstract—In this study, we examine the coupled bending-
torsional dynamic behavior of a cantilever beam carrying
point masses along the span. The eigenfrequencies were
found using the Extended Galerkin Method (EGM) and
validated by the finite element analysis software ANSYS®.
Mainly, two cases were investigated: (i) a beam carrying a
moveable mass along the span and (ii) a beam carrying two
masses (one stationary tip mass and one moving along the
span). Free vibrational analysis was carried out to
demonstrate the effect of the external masses and their
location on the natural frequencies. The results were in
perfect agreement with the finite element analysis. A
coupled bending-torsional behavior is ensured using a kite-
type beam cross section. A validation of the methodology to
the literature is also present for the uncoupled dynamic
behavior. Overall, the results presented in this paper
indicate that the dynamical behavior of beams is highly
dependent on the location and magnitude of the external
mass. A natural frequency decrease was observed as the
mass approached the tip of the beam. Moreover, the
addition of a tip mass to such system was helpful for
dynamic stability. From a practical point of view, this study
will be useful in the design of engineering structures that
carry external stores.
Index Terms—external store, bending torsion coupled
dynamics, Extended Galerkin Method, ANSYS
I. INTRODUCTION
Both in commercial and in military aviation, wings are
designed to carry external stores, such as engines,
missiles, weapons, other defense equipment, or a
Manuscript received July 1, 2018; revised March 23, 2019.
combination of them all. Today’s technology requires
flexibility in the removal and replacement of these stores,
where bombs, rockets, and engines are designed to be
separable from the wing and reattachable in various
combinations, which should be investigated in different
aspects of flight.
From the dynamical viewpoint, both the location and
the weight of an external store on an aircraft’s wing have
profound effects on the behavior of the structure;
therefore, a careful inspection is required to determine
how these parameters affect the dynamical response of
the structure.
So far, several studies have been conducted to
investigate the effect of external stores on the dynamical
characteristics of beam structures with various boundary
conditions. Laura, Pombo, and Susemihl [1] and Parnell
and Cobble [2] determined the natural frequencies of a
fixed-free uniform beam carrying a concentrated tip mass
by solving the transcendental frequency equations
numerically. Gurgoze [3] presented an approximate
formulation to evaluate first two natural frequencies and
mode shapes of beams with an arbitrary number of
concentrated masses. Gokdag and Kopmaz [4]
investigated the coupled bending and torsional vibration
of a beam with a linear spring grounded at the tip mass
and a torsional spring connected at the end. The effect of
the external stores on the dynamical characteristics of
thin-walled composite beams was presented by Na and
Librescu [5] and Librescu and Ohseop [6].
In addition to the dynamic behavior, the aeroelastic
characteristics are significantly affected by these external
stores. Fazelzadeh, Mazidi, and Kalantari [7, 8], Runyan
and Watkins [9], and Xu and Ma [10, 11] studied the
effect on the flutter characteristics of a uniform beam
International Journal of Mechanical Engineering and Robotics Research Vol. 8, No. 3, May 2019
© 2019 Int. J. Mech. Eng. Rob. Res 477doi: 10.18178/ijmerr.8.3.477-482
with an external store. Gern and Librescu [12]
investigated the effect of externally mounted stores on the
thin-walled composite wing aeroelastic behavior. Wang,
Wan, Wu, and Yang [13] modeled and solved the
wing/engine system using the finite element method
(FEM) to capture aeroelastic instabilities. Kashani,
Jayasinghe, and Hashemi [14] solved the dynamic
behavior of a coupled bending-torsional beam that is
under a combined axial end moment load using the
dynamic FEM.
Understanding the dynamical behavior of beams with
multiple point masses carries a great deal of importance
for the design of engineering structures, yet the number
of studies that have been devoted to this area is very
limited. Within this context, in this study, we focus on the
coupled bending-torsional vibrational analysis of a
cantilever beam carrying multiple point masses along the
span. The eigenfrequencies were found using both the
Extended Galerkin Method (EGM) and the finite element
software ANSYS®. Three cases were analyzed: (i) a beam
with a square cross section carrying a tip mass, (ii) a
beam with a kite-type cross section carrying a mass at
different locations along the span, and (iii) a beam with a
kite-type cross section carrying two masses (one
stationary and located at the tip and the other placed at
various locations along the beam span). Overall, the
results presented in this study will be helpful for
understanding the dynamical behavior of beams carrying
external masses. In addition, from a practical point of
view, these results will be utilized in the design of
aeronautical structures, such as an aircraft wing carrying
both missiles and engines.
II. GOVERNING EQUATIONS
Considering a fixed-free beam carrying a point mass of
SM at Sx x , given as in Fig. 1, the equations of
motion featuring bending displacement and angle of twist
can be represented as follows [7]:
'''' ( ) 0S D Smw me EIw M x x w ,
2 '' ( ) 0m S D Smk mew GJ I x x ,
where w and are the bending displacement and angle
of twist of the beam, respectively; m , EI , and GJ are
the mass per unit length, bending rigidity, and torsional
rigidity of the beam, respectively; and Mk is the mass
radius of gyration. e is defined by the distance between
the shear center and the center of gravity of the beam’s
cross section, noting that when 0e , decoupling of
bending and twist motion occurs.
The mass of the store and store mass moment of inertia
are represented by SM and
SI , respectively. Here Sx is
the coordinate of the location of the external mass. In
order to locate the mass at the span, the delta Dirac
function, ( )D x , is used, as suggested by Fazelzadeh,
Mazidi, and Kalantari [7].
xS
external mass:
MS, IS
beam:
m, EI, GJ
Figure 1. Cantilever beam with a point mass.
The displacement of the aforementioned governing
equation is discretized as follows, where ( , )w x t is the
bending displacement and ( , )x t is the twist angle:
( , ) ( ) ( )T
w ww x t N x q t,
(2)
( , ) ( ) ( )Tx t N x q t .
Here, ( )wN x and ( )N x are the shape functions and
( )wq t and ( )q t are the generalized coordinate vectors,
as given below. EGM is applied to solve the natural
frequencies of the beam configurations. The main
advantage of the EGM is its ability to solve the problem
by selecting the weighting functions by only fulfilling the
geometric boundary conditions. This method does not
require natural boundary conditions to be satisfied; hence,
it provides significant advantages for structures with
complex boundary conditions [15].
The shape functions of the governing equations are
chosen to be polynomials to satisfy only the geometric
boundary conditions as follows:
2 3( ) [ ... ]T n
wN x x x x ,
(3) 2( ) [ ... ]T nN x x x x .
Here, T represents the transpose and n is the degree
of polynomial. In order to achieve the desired accuracy,
the polynomial degree is chosen to be 9n .
After obtaining the weak Galerkin form, the governing
equations of motion take the following form:
0 0
0
[ ] 0S
L L
T T T T
w w w w w w
L
T T T T
s w w w w x x w w w w
m q N N q dx me q N N q dx
M q N N q EI q N N q dx
2
0 0
0
[ ] 0S
L L
T T T T
w w
L
T T T T
s x x
mk q N N q dx me q N N q dx
I q N N q GJ q N N q dx
m
Here, the mass matrix, stiffness matrix, and
generalized coordinate vector of the above equations are
defined as follows:
0 0
2
0 0
[ ]
[ ]
[ ]
S
S
L L
T T T
w w s w w x x w
L L
T T T
w s x x
mN N dx M N N meN N dx
M
meN N dx mk N N dx I N N
m
International Journal of Mechanical Engineering and Robotics Research Vol. 8, No. 3, May 2019
© 2019 Int. J. Mech. Eng. Rob. Res 478
(1)
(4)
(5)
0
0
0
[ ]
0
L
T
w w
L
T
EI N N dx
K
GJ N N dx
wqq
q
Assuming a simple harmonic motion, we can substitute
( ) i tq t qe into the equations of motion such that the
generalized coordinates are given in the form
( ) { } , ( ) { }i t i t
w wq t q e q t q e
. Solving the
governing equation of the system, [ ]{ } [ ]{ } 0M q K q
gives the eigenfrequencies of the beam with store masses
in the coupled bending-torsional motion.
III. RESULTS AND DISCUSSION
This section presents the dynamic analysis results of
the cantilever beam carrying external stores. We mainly
carry out a free vibrational analysis for two different
cases in order to show the effect of the external store on
the dynamical behavior of the beams. The first case of the
results is presented for the beam carrying one mass,
whereas the second case is presented for the beam
carrying two separate masses. In both cases, we model
the beam with a kite-type cross section. In addition, the
results in both cases are obtained using the EGM and
compared using the FEM. The results of the FEM are
handled by performing a modal analysis in ANSYS®
software. Lastly, all the frequencies are both tabulated in
tables and plotted in figures, and the correlation between
the EGM and FEM is shown.
Before proceeding to the results section, we will first
validate the accuracy of our dynamic model by
considering a special case from the open literature [1–3].
A beam with a square cross section carrying a tip mass is
chosen, and the obtained results are compared in Table 1.
As the center of gravity coincides with the shear center
[take 0e in Eq. (1)] for a square cross section, the
eigenfrequencies of this beam are decoupled in bending
and torsional modes. Table 1 reveals that our results,
which are provided by the EGM, are in excellent
agreement with [1–3] for varying values of tip mass. As
discussed in a previous study by Wang [13], we similarly
observe that the eigenfrequencies of the beam with a tip
mass decrease as the mass increases.
TABLE I. EIGENFREQUENCIES FOR AN UNCOUPLED BEAM WITH A POINT MASS AT THE TIP.
ω1 (rad/s) ω2 (rad/s) ω3 (rad/s)
MS Present [3] [1, 2] Error% Present [3] [1, 2] Error% Present [1, 2] Error%
0.2 2.61 2.61 2.61 0 18.21 18.38 18.21 0 53.56 53.56 0
1 1.56 1.56 1.56 0 16.25 16.56 16.25 0 50.9 50.9 0
2 1.16 1.16 1.16 0 15.87 16.19 15.86 0.06 50.45 50.45 0
5 0.76 0.76 0.76 0 15.6 15.95 15.6 0 50.17 50.16 0.02
10 0.54 0.54 0.54 0 15.51 15.87 15.51 0 50.07 50.06 0.02
100 0.17 0.17 0.17 0 15.43 15.79 15.43 0 49.98 49.97 0.02
TABLE II. COUPLED BENDING-TORSIONAL BEAM AND MASS PROPERTIES.
Area A 26 m2
Mass per unit length m 7.02E − 05 kg/m
Poisson’s ratio ν 0.3 —
Young’s modulus E 7.00E + 04 N/m2
Shear center (y, z) SC (1.468, 0) m
Centroid (y, z) CG (2.256, 0) m
External mass M 0.01 kg
Length of beam l 200 m
International Journal of Mechanical Engineering and Robotics Research Vol. 8, No. 3, May 2019
© 2019 Int. J. Mech. Eng. Rob. Res 479
TABLE III. EIGENFREQUENCIES FOR A COUPLED BENDING-TORSIONAL BEAM WITH A MASS.
ξ = L/8 ξ = L/4 ξ = L/2 ξ = 3L/4 ξ = L
Mode EGM FEM Error% EGM FEM Error% EGM FEM Error% EGM FEM Error% EGM FEM Error%
ω1 1.93 1.93 0.02 1.91 1.91 0.02 1.67 1.67 0.02 1.29 1.29 0.02 0.98 0.98 0.01
ω2 5.80 5.80 0.13 5.73 5.72 0.14 5.02 5.01 0.17 3.89 3.88 0.14 2.94 2.93 0.09
ω3 11.77 11.75 0.17 9.66 9.64 0.20 8.36 8.35 0.15 11.97 11.96 0.13 9.09 9.08 0.10
ω4 28.46 28.26 0.69 23.86 23.75 0.44 25.12 24.88 0.93 27.87 27.81 0.22 27.19 27.10 0.36
ω5 35.30 34.92 1.08 28.99 28.59 1.39 33.91 33.81 0.31 35.91 35.59 0.91 28.16 28.09 0.26
ω6 49.98 49.47 1.03 56.81 56.44 0.65 53.29 52.78 0.94 57.75 57.58 0.30 57.76 57.48 0.48
ω7 70.85 70.85 0.00 69.19 69.83 −0.92 70.82 70.81 0.01 70.40 70.84 −0.63 70.83 70.82 0.02
TABLE IV. EIGENFREQUENCIES FOR AN UNCOUPLED BENDING-TORSIONAL BEAM WITH TWO MASSES.
ξ = L/8 ξ = L/4 ξ = L/2 ξ = 3L/4
Mode EGM FEM
Error% EGM FEM
Error% EGM FEM
Error% EGM FEM
Error%
ω1
0.35 0.35 0.01 0.35 0.35 0.01 0.35 0.35 0.01 0.34 0.34 0.01
ω2
1.06 1.05 0.08 1.05 1.05 0.07 1.05 1.05 0.08 1.04 1.03 0.08
ω3
8.42 8.41 0.08 7.36 7.35 0.14 5.41 5.40 0.12 5.95 5.95 0.10
ω4
24.23 24.11 0.48 18.95 18.88 0.36 16.23 16.12 0.71 17.87 17.76 0.60
ω5
25.19 25.09 0.40 22.07 21.90 0.78 26.24 26.21 0.14 21.55 21.48 0.29
A. Coupled Bending-Torsional Beam Carrying a Mass
In this first set, we demonstrate the results of the
dynamic analysis of a cantilever beam carrying one mass.
By altering the span location of the mass, we aim to
investigate the effect of the location of the external mass
on the frequencies of the beam. The material and
geometric properties of the beam used in this section are
listed in Table II.
We consider the beam with an external mass, SM ,
located at a distance from the fixed end. The cross
section of the beam is chosen as a kite-type section,
whose sketch is given in Fig. 3.
The condition of ξ = 0 refers to the fixed-free beam
natural frequency itself. The eigenfrequency results
displayed in Fig. 4 and tabulated in Table 3 are compared
with the results of the finite element analysis using
ANSYS® software. The beam structure is divided into
200 elements using linear, six-degree-of-freedom
BEAM188 elements. The results are plotted up to the
seventh natural frequency where the torsional mode is
observed. The results of both methods overlap with each
other, even for higher-order modes. The bending
frequencies decrease as the mass gets closer to the tip,
which can be clearly observed at the first two modes.
Higher-order modes do not show a stable trend when
compared to the lower-order modes.
International Journal of Mechanical Engineering and Robotics Research Vol. 8, No. 3, May 2019
© 2019 Int. J. Mech. Eng. Rob. Res 480
a
L
MS
Figure 2. Uncoupled bending-torsional beam with a point mass.
MS
xS
Figure 3. Coupled bending-torsional beam.
Figure 4. Natural frequencies of a coupled bending-torsional beam with
a mass (refer to Fig. 3 for configuration).
B. Coupled Bending-Torsional Beam with Multiple
Masses
In this section, we present the results of the dynamical
analysis of a coupled bending-torsional beam carrying
two masses. The material properties of the beam are
identical to those of the previous numerical example,
which is defined in Table II. The sketch of the cross
section of the beam and the beam geometry are given in
Figs. 3(b) and 5, respectively. The external mass, 1SM , is
chosen as stationary at the tip of the beam, whereas the
point mass, 2SM , is chosen to be placed in various
locations along the beam span.
An analysis similar to that in the previous case is
conducted, and the obtained results are compared with the
ones of the FEM and given in Table IV. As seen from
Table IV and Fig. 6, the natural frequency results are
provided with subtle changes as the location of the point
mass is altered. As expected, such behavior vanishes at
higher-order modes. Additionally, including a second
mass resulted in a decrease in the frequencies, regardless
of the location of the second mass.
MS2
xS
L
MS1
Figure 5. Coupled bending-torsional beam with two masses.
Figure 6. Natural frequencies of a coupled bending-torsional beam with
two masses.
IV. CONCLUSIONS
In this study, we focused on the dynamical analysis of
the coupled bending-torsional behavior of a cantilever
beam carrying external masses. The eigenfrequencies
were found using both EGM and the finite element
software ANSYS®. The obtained results of both methods
were compared with each other. We found that the EGM
provides a high accuracy with rapid convergence and
maintains this ability for even higher-order modes.
Two different beam-mass configurations were
presented and analyzed in order to demonstrate the
effects of external masses on the dynamical
characteristics of the beam. We initially validated our
results using EGM for a beam with a square cross section
with a tip mass against the open literature, showing a very
good agreement. In the first beam-mass configuration, we
placed a mass at different span locations and performed
free vibrational analyses to obtain the natural frequencies
of the beam. The results showed that as the mass gets
closer to the tip, a larger decrease in the natural frequency
is observed. In the second configuration, the dynamical
results of a beam carrying two masses were presented.
For this one, the first mass was kept stationary at the tip
of the beam and the second mass was placed at various
locations along the span. The results of this case similarly
showed that the tip mass provides a more stable
International Journal of Mechanical Engineering and Robotics Research Vol. 8, No. 3, May 2019
© 2019 Int. J. Mech. Eng. Rob. Res 481
dynamical behavior, regardless of the different
placements of the second mass. In addition, we observed
that higher-order modes do not show a notable trend and
we see that natural frequencies vary as the mass take
different locations along the span.
REFERENCES
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[4] H. Gokdag, O. Kopmaz, “Coupled bending and torsional vibration of a beam with in-span and tip attachments,” Journal of
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292-305, 2013.
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Alev Kacar Aksongur was born in Istanbul, Turkey, in 1983. She
graduated from Istanbul Technical University (Department of Astronautical Engineering), in 2007, Istanbul, Turkey. She obtained her
M.Sc. degree from the same department in 2009, where her main study
area was active vibration control of piezoelectric materials. She is a Ph.D. candidate at the Aeronautical and Astronautical Engineering
Interdisciplinary Department, where her major focus area is the
aeroelastic behavior of composite beam structures. Alev is currently working as a Design Subsection Manager of the LM2500 CDN module
in GE Aviation, Kocaeli, Turkey, where she has worked in various
positions since 2010. Before that, she was a researcher from 2007 to 2010 at Istanbul Technical University. Her current research area is the
aeroelasticity and flutter behavior of thin-walled composite beams
carrying external stores. Her recent publications include Flexural-Torsional Vibration Analysis of Thin-Walled Composite Wings
Carrying External Stores, ASME International Mechanical Engineering
Congress & Exposition, November 2014, Montreal, Canada
.
International Journal of Mechanical Engineering and Robotics Research Vol. 8, No. 3, May 2019
© 2019 Int. J. Mech. Eng. Rob. Res 482